$\dfrac{ c + 3d }{ -9 } = \dfrac{ -4c - e }{ -6 }$ Solve for $c$.
Solution: Multiply both sides by the left denominator. $\dfrac{ c + 3d }{ -{9} } = \dfrac{ -4c - e }{ -6 }$ $-{9} \cdot \dfrac{ c + 3d }{ -{9} } = -{9} \cdot \dfrac{ -4c - e }{ -6 }$ $c + 3d = -{9} \cdot \dfrac { -4c - e }{ -6 }$ Multiply both sides by the right denominator. $c + 3d = -9 \cdot \dfrac{ -4c - e }{ -{6} }$ $-{6} \cdot \left( c + 3d \right) = -{6} \cdot -9 \cdot \dfrac{ -4c - e }{ -{6} }$ $-{6} \cdot \left( c + 3d \right) = -9 \cdot \left( -4c - e \right)$ Distribute both sides $-{6} \cdot \left( c + 3d \right) = -{9} \cdot \left( -4c - e \right)$ $-{6}c - {18}d = {36}c + {9}e$ Combine $c$ terms on the left. $-{6c} - 18d = {36c} + 9e$ $-{42c} - 18d = 9e$ Move the $d$ term to the right. $-42c - {18d} = 9e$ $-42c = 9e + {18d}$ Isolate $c$ by dividing both sides by its coefficient. $-{42}c = 9e + 18d$ $c = \dfrac{ 9e + 18d }{ -{42} }$ All of these terms are divisible by $3$ Divide by the common factor and swap signs so the denominator isn't negative. $c = \dfrac{ -{3}e - {6}d }{ {14} }$